Problem 92 - Intermediate Algebra

Question

Simplify each expression. $$3 T-2(t-T)+t$$

Step-by-Step Solution

Answer

The expression simplifies to \(5T - t\).

Step 1: Expand the Expression

Let's begin by expanding the expression. The given expression is \(3T - 2(t - T) + t\). Distribute the \(-2\) across the terms inside the parenthesis. This gives us \(3T - 2t + 2T + t\).

Step 2: Combine Like Terms

Next, combine the like terms from the expanded expression \(3T - 2t + 2T + t\). The terms \(3T\) and \(2T\) can be combined to get \(5T\). Similarly, the terms \(-2t\) and \(+t\) simplify to \(-t\).

Step 3: Final Expression

The simplified form of the expression after combining like terms is \(5T - t\).

Key Concepts

Expanding Expressions

Expanding expressions is an important preliminary step in algebra. This process involves removing parentheses by distributing any coefficients across the terms inside the parentheses. In algebraic expressions such as \(a(b + c)\), we multiply \(a\) by both \(b\) and \(c\) to get \(ab + ac\). For our specific problem, the expression is \(3T - 2(t - T) + t\). The goal here is to distribute the \(-2\) through the terms inside the parentheses.

• Multiply \(-2\) by \(t\) to get \(-2t\).
• Multiply \(-2\) by \(-T\) to get \(+2T\), because multiplying two negatives results in a positive.

Putting it all together, the expression expands to \(3T - 2t + 2T + t\). This step ensures that every term is accounted for separately and can make combining terms more straightforward.

Combining Like Terms

Once an expression is expanded, the next step is to simplify it by combining like terms. "Like terms" are terms that contain the same variable raised to the same power, but they can have different coefficients. For instance, in the expression \(3T - 2t + 2T + t\), we can identify the following like terms:

• Terms with \(T\): \(3T\) and \(2T\)
• Terms with \(t\): \(-2t\) and \(+t\)

To combine these terms:

• Add \(3T\) and \(2T\) to get \(5T\).
• Add \(-2t\) and \(+t\) to get \(-t\) (since \(1t - 2t = -t\)).

By combining like terms, the expression now condenses to \(5T - t\), making it simpler and more manageable.

Simplifying Expressions

The final goal in algebra is often to simplify expressions as much as possible. This means reducing the expression to its simplest form without changing its value. Simplifying involves both expanding expressions and combining like terms, as seen in the previous steps. Our task was to simplify \(3T - 2(t - T) + t\), which through expansion and combining becomes \(5T - t\).Simplifying can be viewed as organizing and reducing the expression to its "cleanest" state. The final expression, \(5T - t\), doesn't have any like terms left to combine and is the simplest version of the original expression.

• This makes it easier to solve if it were to be part of an equation.
• It also makes it more comprehensible and aesthetically pleasing.

When you simplify, you ensure that your math work is as clear and correct as possible, which is crucial in mathematics!